Oscillating Rim Hook Tableaux and Colored Matchings

TL;DR

This paper establishes a novel combinatorial correspondence between oscillating m-rim hook tableaux and m-colored matchings, extending classical bijections and revealing new connections with Catalan numbers and Dyck path packings.

Contribution

It introduces a new bijection between oscillating m-rim hook tableaux and m-colored matchings, generalizing known results and linking to Dyck path packings.

Findings

Bijection between oscillating m-rim hook tableaux and m-colored matchings.

Special case for domino tableaux relating to noncrossing 2-colored matchings.

Dyck path packings counted by the product of two Catalan numbers.

Abstract

Motivated by the question of finding a type B analogue of the bijection between oscillating tableaux and matchings, we find a correspondence between oscillating m-rim hook tableaux and m-colored matchings, where m is a positive integer. An oscillating m-rim hook tableau is defined as a sequence $(\lambda^0,\lambda^1,...,\lambda^{2n})$ of Young diagrams starting with the empty shape and ending with the empty shape such that $\lambda^{i}$ is obtained from $\lambda^{i-1}$ by adding an m-rim hook or by deleting an m-rim hook. Our bijection relies on the generalized Schensted algorithm due to White. An oscillating 2-rim hook tableau is also called an oscillating domino tableau. When we restrict our attention to two column oscillating domino tableaux of length 2n, we are led to a bijection between such tableaux and noncrossing 2-colored matchings on $\{1, 2,..., 2n\}$, which are counted by the product $C_nC_{n+1}$ of two consecutive Catalan numbers. A 2-colored matching is noncrossing if there are no two arcs of the same color that are intersecting. We show that oscillating domino tableaux with at most two columns are in one-to-one correspondence with Dyck path packings. A Dyck path packing of length 2n is a pair (D, E), where D is a Dyck path of length 2n, and E is a dispersed Dyck path of length 2n that is weakly covered by D. So we deduce that Dyck path packings of length 2n are counted by $C_nC_{n+1}$.